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Properties

Label 2-260-260.103-c1-0-5
Degree $2$
Conductor $260$
Sign $-0.581 + 0.813i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.358 + 1.36i)2-s + (−1.80 + 1.80i)3-s + (−1.74 + 0.980i)4-s + (−1.12 + 1.92i)5-s + (−3.11 − 1.82i)6-s + (−0.163 + 0.163i)7-s + (−1.96 − 2.03i)8-s − 3.52i·9-s + (−3.04 − 0.854i)10-s + 5.50·11-s + (1.37 − 4.91i)12-s + (0.177 − 3.60i)13-s + (−0.282 − 0.165i)14-s + (−1.44 − 5.52i)15-s + (2.07 − 3.41i)16-s + (−5.33 + 5.33i)17-s + ⋯
L(s)  = 1  + (0.253 + 0.967i)2-s + (−1.04 + 1.04i)3-s + (−0.871 + 0.490i)4-s + (−0.505 + 0.862i)5-s + (−1.27 − 0.744i)6-s + (−0.0619 + 0.0619i)7-s + (−0.695 − 0.718i)8-s − 1.17i·9-s + (−0.962 − 0.270i)10-s + 1.65·11-s + (0.397 − 1.42i)12-s + (0.0491 − 0.998i)13-s + (−0.0755 − 0.0441i)14-s + (−0.372 − 1.42i)15-s + (0.519 − 0.854i)16-s + (−1.29 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ -0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.283812 - 0.551350i\)
\(L(\frac12)\) \(\approx\) \(0.283812 - 0.551350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.358 - 1.36i)T \)
5 \( 1 + (1.12 - 1.92i)T \)
13 \( 1 + (-0.177 + 3.60i)T \)
good3 \( 1 + (1.80 - 1.80i)T - 3iT^{2} \)
7 \( 1 + (0.163 - 0.163i)T - 7iT^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
17 \( 1 + (5.33 - 5.33i)T - 17iT^{2} \)
19 \( 1 - 3.95iT - 19T^{2} \)
23 \( 1 + (3.90 - 3.90i)T - 23iT^{2} \)
29 \( 1 + 2.85iT - 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 + (-1.81 - 1.81i)T + 37iT^{2} \)
41 \( 1 - 1.07iT - 41T^{2} \)
43 \( 1 + (-2.39 + 2.39i)T - 43iT^{2} \)
47 \( 1 + (-2.94 + 2.94i)T - 47iT^{2} \)
53 \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \)
59 \( 1 - 3.40iT - 59T^{2} \)
61 \( 1 + 3.68T + 61T^{2} \)
67 \( 1 + (2.55 - 2.55i)T - 67iT^{2} \)
71 \( 1 + 0.844T + 71T^{2} \)
73 \( 1 + (3.98 - 3.98i)T - 73iT^{2} \)
79 \( 1 + 6.47T + 79T^{2} \)
83 \( 1 + (-8.45 - 8.45i)T + 83iT^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 + (6.74 + 6.74i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.41611024424653012429621830437, −11.63012508719421664987812419971, −10.68741971722304567111985924074, −9.838990796798055232869811129694, −8.718928257628156528017561468887, −7.50649929164955897194848640150, −6.25827762528884504873645316235, −5.83895313809017119681988302429, −4.21316260788865487206024172982, −3.76034392868132845405795451645, 0.52423151066495061203374400620, 1.87509793336210071274573035000, 4.06527366405925612479275995606, 4.93566248842870550049074945148, 6.29856973311551073815100006064, 7.13678137084632868365130629567, 8.843336399533428274530748055695, 9.308365047271271501405609360168, 11.02770113900135205174436955713, 11.68449034076409501542012466557

Graph of the $Z$-function along the critical line